A current through a wire varies with time, as $I=αt + β t^2$ where $α = 10 A s^{-1}$ and $β = 4 A s^{-2}$. How much charge will cross through a section of the wire in 12 s?

3024 C

The correct answer is Option (4) → 3024 C

Given:
$I = \alpha t + \beta t^2$
$\alpha = 10\,A\,s^{-1}$, $\beta = 4\,A\,s^{-2}$, $t = 12\,s$

Charge passed through the wire:

$Q = \int_0^{t} I\,dt = \int_0^{12} (\alpha t + \beta t^2)\,dt$

$Q = \alpha \int_0^{12} t\,dt + \beta \int_0^{12} t^2\,dt$

$Q = \alpha \left[\frac{t^2}{2}\right]_0^{12} + \beta \left[\frac{t^3}{3}\right]_0^{12}$

$Q = 10 \left(\frac{12^2}{2}\right) + 4 \left(\frac{12^3}{3}\right)$

$Q = 10(72) + 4(576)$

$Q = 720 + 2304 = 3024\,C$

∴ Total charge = 3024 C